Rolle's theorem is an important theorem among the class of results regarding the value of the derivative on an interval.
Let be continous on and differentiable on
Then such that
The result is trivial for the case . Hence, let us assume that is a non-constant function.
Let and Without loss of generality, we can assume that
By the Maximum-minimum theorem, such that
Assume if possible
Hence, such that
Thus we have that if , contradicting the assumption that is a maximum.
Similarly we can show that leads to contradiction.